Development and stability analysis of the inverse Lax−Wendroff boundary treatment for central compact schemes
Vilar, François1 ; Shu, Chi-Wang1
1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 39-67.

In this paper, we generalize the so-called inverse Lax−Wendroff boundary treatment [S. Tan and C.-W. Shu, J. Comput. Phys. 229 (2010) 8144–8166] for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes [X. Liu, S. Zhang, H. Zhang and C.-W. Shu, J. Comput. Phys. 248 (2013) 235–256]. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory [B. Gustafsson, H.-O. Kreiss and A. Sundström, Math. Comput. 26 (1972) 649–686] is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge−Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems.

Reçu le :
DOI : 10.1051/m2an/2014024
Classification : 65M12
Mots-clés : Central compact schemes, initial boundary value problem, inverse Lax−Wendroff, extrapolation, G-K-S theory, eigenvalue spectrum
Vilar, François 1 ; Shu, Chi-Wang 1

1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. 
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     title = {Development and stability analysis of the inverse {Lax\ensuremath{-}Wendroff} boundary treatment for central compact schemes},
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     publisher = {EDP-Sciences},
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Vilar, François; Shu, Chi-Wang. Development and stability analysis of the inverse Lax−Wendroff boundary treatment for central compact schemes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 39-67. doi : 10.1051/m2an/2014024. http://archive.numdam.org/articles/10.1051/m2an/2014024/

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